Abstract
We investigate the dynamics responsible for generating the potential of the η′, the (would-be) Goldstone boson associated with the anomalous axial U(1) symmetry of QCD. The standard lore posits that pure QCD dynamics generates a confining potential with a branched structure as a function of the θ angle, and that this same potential largely determines the properties of the η′ once fermions are included. Here we test this picture by examining a supersymmetric extension of QCD with a small amount of supersymmetry breaking generated via anomaly mediation. For pure SU(N) QCD without flavors, we verify that there are N branches generated by gaugino condensation. Once quarks are introduced, the flavor effects qualitatively change the strong dynamics of the pure theory. For F flavors we find |N − F| branches, whose dynamical origin is gaugino condensation in the unbroken subgroup for F < N – 1, and in the dual gauge group for F > N + 1. For the special cases of F = N – 1, N, N + 1 we find no branches and the entire potential is consistent with being a one-instanton effect. The number of branches is a simple consequence of the selection rules of an anomalous U(1)R symmetry. We find that the η′ mass does not vanish in the large N limit for fixed F/N, since the anomaly is non-vanishing. The same dynamics that is responsible for the η′ potential is also responsible for the axion potential. We present a simple derivation of the axion mass formula for an arbitrary number of flavors.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Witten, Instantons, the Quark Model, and the 1/n Expansion, Nucl. Phys. B 149 (1979) 285 [INSPIRE].
P. Di Vecchia and G. Veneziano, Chiral Dynamics in the Large n Limit, Nucl. Phys. B 171 (1980) 253 [INSPIRE].
K. Kawarabayashi and N. Ohta, The Problem of η in the Large N Limit: Effective Lagrangian Approach, Nucl. Phys. B 175 (1980) 477 [INSPIRE].
N. Ohta, Vacuum Structure and Chiral Charge Quantization in the Large N Limit, Prog. Theor. Phys. 66 (1981) 1408 [Erratum ibid. 67 (1982) 993] [INSPIRE].
L. Randall and R. Sundrum, Out of this world supersymmetry breaking, Nucl. Phys. B 557 (1999) 79 [hep-th/9810155] [INSPIRE].
G.F. Giudice, M.A. Luty, H. Murayama and R. Rattazzi, Gaugino mass without singlets, JHEP 12 (1998) 027 [hep-ph/9810442] [INSPIRE].
N. Arkani-Hamed, G.F. Giudice, M.A. Luty and R. Rattazzi, Supersymmetry breaking loops from analytic continuation into superspace, Phys. Rev. D 58 (1998) 115005 [hep-ph/9803290] [INSPIRE].
M.A. Luty and R. Rattazzi, Soft supersymmetry breaking in deformed moduli spaces, conformal theories, and N = 2 Yang-Mills theory, JHEP 11 (1999) 001 [hep-th/9908085] [INSPIRE].
A. Pomarol and R. Rattazzi, Sparticle masses from the superconformal anomaly, JHEP 05 (1999) 013 [hep-ph/9903448] [INSPIRE].
I. Affleck, M. Dine and N. Seiberg, Dynamical Supersymmetry Breaking in Supersymmetric QCD, Nucl. Phys. B 241 (1984) 493 [INSPIRE].
N. Seiberg, Exact results on the space of vacua of four-dimensional SUSY gauge theories, Phys. Rev. D 49 (1994) 6857 [hep-th/9402044] [INSPIRE].
N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].
K.A. Intriligator and N. Seiberg, Lectures on supersymmetric gauge theories and electric-magnetic duality, Nucl. Phys. B Proc. Suppl. 45BC (1996) 1 [hep-th/9509066] [INSPIRE].
H. Murayama, Some Exact Results in QCD-like Theories, Phys. Rev. Lett. 126 (2021) 251601 [arXiv:2104.01179] [INSPIRE].
C. Csáki et al., Guide to anomaly-mediated supersymmetry-breaking QCD, Phys. Rev. D 107 (2023) 054015 [arXiv:2212.03260] [INSPIRE].
C. Csáki, H. Murayama and O. Telem, More exact results on chiral gauge theories: The case of the symmetric tensor, Phys. Rev. D 105 (2022) 045007 [arXiv:2105.03444] [INSPIRE].
C. Csáki, A. Gomes, H. Murayama and O. Telem, Demonstration of Confinement and Chiral Symmetry Breaking in SO(Nc) Gauge Theories, Phys. Rev. Lett. 127 (2021) 251602 [arXiv:2106.10288] [INSPIRE].
C. Csáki, H. Murayama and O. Telem, Some exact results in chiral gauge theories, Phys. Rev. D 104 (2021) 065018 [arXiv:2104.10171] [INSPIRE].
C. Csáki, A. Gomes, H. Murayama and O. Telem, Phases of nonsupersymmetric gauge theories: The SO(Nc) case study, Phys. Rev. D 104 (2021) 114018 [arXiv:2107.02813] [INSPIRE].
H. Murayama, B. Noether and D.R. Varier, Broken Conformal Window, arXiv:2111.09690 [INSPIRE].
Y. Bai and D. Stolarski, Phases of confining SU(5) chiral gauge theory with three generations, JHEP 03 (2022) 113 [arXiv:2111.11214] [INSPIRE].
D. Kondo, H. Murayama and C. Sylber, Dynamics of Simplest Chiral Gauge Theories, arXiv:2209.09287 [INSPIRE].
A. Luzio and L.-X. Xu, On the derivation of chiral symmetry breaking in QCD-like theories and S-confining theories, JHEP 08 (2022) 016 [arXiv:2202.01239] [INSPIRE].
L. Ciambriello, R. Contino and L.-X. Xu, On the Proof of Chiral Symmetry Breaking from Anomaly Matching in QCD-like Theories, arXiv:2212.02930 [INSPIRE].
D. Gaiotto, Z. Komargodski and N. Seiberg, Time-reversal breaking in QCD4, walls, and dualities in 2 + 1 dimensions, JHEP 01 (2018) 110 [arXiv:1708.06806] [INSPIRE].
M. Dine, P. Draper, L. Stephenson-Haskins and D. Xu, θ and the η′ in Large N Supersymmetric QCD, JHEP 05 (2017) 122 [arXiv:1612.05770] [INSPIRE].
D. Davies, M. Dine and B.V. Lehmann, Light Quarks at Large N, arXiv:2201.05719 [INSPIRE].
G. ’t Hooft, Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle, Phys. Rev. D 14 (1976) 3432 [Erratum ibid. 18 (1978) 2199] [INSPIRE].
G. ’t Hooft, How Instantons Solve the U(1) Problem, Phys. Rept. 142 (1986) 357 [INSPIRE].
K. Fujikawa, Path Integral Measure for Gauge Invariant Fermion Theories, Phys. Rev. Lett. 42 (1979) 1195 [INSPIRE].
K. Fujikawa, Path Integral for Gauge Theories with Fermions, Phys. Rev. D 21 (1980) 2848 [Erratum ibid. 22 (1980) 1499] [INSPIRE].
G. Veneziano, U(1) Without Instantons, Nucl. Phys. B 159 (1979) 213 [INSPIRE].
E. Witten, Large N Chiral Dynamics, Annals Phys. 128 (1980) 363 [INSPIRE].
E. Witten, Current Algebra Theorems for the U(1) Goldstone Boson, Nucl. Phys. B 156 (1979) 269 [INSPIRE].
B. Holdom and M.E. Peskin, Raising the Axion Mass, Nucl. Phys. B 208 (1982) 397 [INSPIRE].
B. Holdom, Strong QCD at High-energies and a Heavy Axion, Phys. Lett. B 154 (1985) 316 [Erratum ibid. 156 (1985) 452] [INSPIRE].
J.M. Flynn and L. Randall, A Computation of the Small Instanton Contribution to the Axion Potential, Nucl. Phys. B 293 (1987) 731 [INSPIRE].
P. Agrawal and K. Howe, Factoring the Strong CP Problem, JHEP 12 (2018) 029 [arXiv:1710.04213] [INSPIRE].
C. Csáki, M. Ruhdorfer and Y. Shirman, UV Sensitivity of the Axion Mass from Instantons in Partially Broken Gauge Groups, JHEP 04 (2020) 031 [arXiv:1912.02197] [INSPIRE].
T. Gherghetta, V.V. Khoze, A. Pomarol and Y. Shirman, The Axion Mass from 5D Small Instantons, JHEP 03 (2020) 063 [arXiv:2001.05610] [INSPIRE].
L. Randall, R. Rattazzi and E.V. Shuryak, Implication of exact SUSY gauge couplings for QCD, Phys. Rev. D 59 (1999) 035005 [hep-ph/9803258] [INSPIRE].
N.J. Evans, S.D.H. Hsu and M. Schwetz, Exact results in softly broken supersymmetric models, Phys. Lett. B 355 (1995) 475 [hep-th/9503186] [INSPIRE].
L. Alvarez-Gaume, J. Distler, C. Kounnas and M. Marino, Softly broken N = 2 QCD, Int. J. Mod. Phys. A 11 (1996) 4745 [hep-th/9604004] [INSPIRE].
K. Konishi, Confinement, supersymmetry breaking and theta parameter dependence in the Seiberg-Witten model, Phys. Lett. B 392 (1997) 101 [hep-th/9609021] [INSPIRE].
H.-C. Cheng and Y. Shadmi, Duality in the presence of supersymmetry breaking, Nucl. Phys. B 531 (1998) 125 [hep-th/9801146] [INSPIRE].
N. Arkani-Hamed and R. Rattazzi, Exact results for nonholomorphic masses in softly broken supersymmetric gauge theories, Phys. Lett. B 454 (1999) 290 [hep-th/9804068] [INSPIRE].
S. Abel, M. Buican and Z. Komargodski, Mapping Anomalous Currents in Supersymmetric Dualities, Phys. Rev. D 84 (2011) 045005 [arXiv:1105.2885] [INSPIRE].
C. Córdova and T.T. Dumitrescu, Candidate Phases for SU(2) Adjoint QCD4 with Two Flavors from \( \mathcal{N} \) = 2 Supersymmetric Yang-Mills Theory, arXiv:1806.09592 [INSPIRE].
I. Affleck, M. Dine and N. Seiberg, Dynamical Supersymmetry Breaking in Four-Dimensions and Its Phenomenological Implications, Nucl. Phys. B 256 (1985) 557 [INSPIRE].
S.F. Cordes, The Instanton Induced Superpotential in Supersymmetric QCD, Nucl. Phys. B 273 (1986) 629 [INSPIRE].
M.A. Shifman and A.I. Vainshtein, On Gluino Condensation in Supersymmetric Gauge Theories. SU(N) and O(N) Groups, Sov. Phys. JETP 66 (1987) 1100 [INSPIRE].
D. Finnell and P. Pouliot, Instanton calculations versus exact results in four-dimensional SUSY gauge theories, Nucl. Phys. B 453 (1995) 225 [hep-th/9503115] [INSPIRE].
A.C. Davis, M. Dine and N. Seiberg, The Massless Limit of Supersymmetric QCD, Phys. Lett. B 125 (1983) 487 [INSPIRE].
E. Witten, Theta dependence in the large N limit of four-dimensional gauge theories, Phys. Rev. Lett. 81 (1998) 2862 [hep-th/9807109] [INSPIRE].
D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, Time Reversal, and Temperature, JHEP 05 (2017) 091 [arXiv:1703.00501] [INSPIRE].
K.A. Intriligator, N. Seiberg and D. Shih, Dynamical SUSY breaking in meta-stable vacua, JHEP 04 (2006) 021 [hep-th/0602239] [INSPIRE].
A. Armoni, M. Shifman and G. Veneziano, QCD quark condensate from SUSY and the orientifold large N expansion, Phys. Lett. B 579 (2004) 384 [hep-th/0309013] [INSPIRE].
N. Arkani-Hamed and H. Murayama, Holomorphy, rescaling anomalies and exact beta functions in supersymmetric gauge theories, JHEP 06 (2000) 030 [hep-th/9707133] [INSPIRE].
M. Dine et al., Supersymmetric QCD: Exact Results and Strong Coupling, JHEP 05 (2011) 061 [arXiv:1104.0461] [INSPIRE].
M.A. Shifman and A.I. Vainshtein, Solution of the Anomaly Puzzle in SUSY Gauge Theories and the Wilson Operator Expansion, Nucl. Phys. B 277 (1986) 456 [INSPIRE].
S.P. Martin and M.T. Vaughn, Two loop renormalization group equations for soft supersymmetry breaking couplings, Phys. Rev. D 50 (1994) 2282 [Erratum ibid. 78 (2008) 039903] [hep-ph/9311340] [INSPIRE].
Acknowledgments
We thank Adi Armoni, Michael Dine, Hyung Do Kim, Zohar Komargodsky, Hitoshi Murayama, Ofri Telem and Lian-Tao Wang for useful comments and discussions. CC thanks the Tata Institute for Fundamental Research in Mumbai and the Kavli IPMU in Tokyo for its hospitality while this paper was prepared. RTD acknowledges the hospitality of the Ecole de Physique des Houches while part of this paper was completed. CC and MR are supported in part by the NSF grant PHY-2014071. MR is also supported by a Feodor-Lynen Research Fellowship awarded by the Humboldt Foundation. CC is also supported in part by a Simons Foundation Sabbatical Fellowship. CC and EK are funded in part by the US-Israeli BSF grant 2016153.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2307.04809
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Csáki, C., D’Agnolo, R.T., Gupta, R.S. et al. On the dynamical origin of the η′ potential and the axion mass. J. High Energ. Phys. 2023, 139 (2023). https://doi.org/10.1007/JHEP10(2023)139
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2023)139